Detectives said that on three separate occasions in the past few days, foreign visitors had fallen victim to the scam and ended up having money taken from them by the fraudsters.
One woman had cash stolen from her purse after she was approached by a man pretending to be a tourist in St Andrews Square before two men, who told her they were police officers, appeared and spoke to her.
Detective Sergeant Willie Falconer, of Police Scotland, said: "The so-called police officers then asked to see wallets on the pretext of checking for counterfeit currency and identification. They took some cash from the wallets, appeared to sniff the money, replace it and then handed the wallets back. But after the men left, the woman discovered that there was £140 missing.
"All of the incidents are very alike and there is clearly a group of thieves using the same tactic to try and make off with cash from unsuspecting tourists."
Mr Falconer said the crimes, which have also taken place in Princes Street and Johnston Terrace, were being carried out "so quickly that the victims are being duped into thinking the cash is still in their wallets or purses". He added: "We take crimes of theft and impersonating police officers very seriously and will do our utmost to catch the culprits."